Neural quantum embedding via deterministic quantum computation with one qubit

Daniel K. Park; Dawei Lu; Hongfeng Liu; Keyi Huang; Liangyu Che; Shitao Zhang; Tak Hur; Xiangyu Wang; Xinfang Nie; Xinyue Long

REVIEW 4 major objections 4 minor 60 references

Neural quantum embedding via deterministic quantum computation with one qubit

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read One clean qubit can train a quantum data embedding, lifting MNIST '0/1' classification on NMR hardware from 54% to 98%.

desk verdict Solid DQC1-on-NMR proof of principle, but the headline classification gain is confounded by the classical neural network and a missing baseline. read the letter →

arxiv 2501.15359 v1 pith:ZVWKKPKY submitted 2025-01-26 quant-ph

Hongfeng Liu , Tak Hur , Shitao Zhang , Liangyu Che , Xinyue Long , Xiangyu Wang , Keyi Huang , Yu-ang Fan

show 5 more authors

Yuxuan Zheng Yufang Feng Xinfang Nie Daniel K. Park Dawei Lu

This is my paper · ORCID

classification quant-ph MSC 81P68 PACS 03.67.Lx76.60.-k

keywords neuralquantumembeddingDQC1onecleanqubitmachinelearningdataNMRprocessortracedistanceHilbert–Schmidtinnerproduct

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's central claim is that the choice of quantum data embedding—normally a fixed, hand-picked feature map—can instead be learned, and learned with very little quantum hardware. The authors train a classical neural network whose outputs set the angles of a $ZZ$-feature map, using the DQC1 protocol to estimate the overlap between two embedding circuits from a single readout on one clean qubit. On a four-qubit NMR processor encoding handwritten '0' and '1' images, the trained embedding increases the trace distance between the two classes and lifts classification accuracy from 54% (near random guessing) to 98% for the same shallow classifier. If this holds, ensemble-style quantum machines usually set aside as too weak for machine learning become practical trainers for embeddings that later run on other hardware, and the field's bottleneck shifts from the classifier circuit to the data-loading stage.

What carries the argument

The load-bearing object is the DQC1 overlap estimator: starting from one pure probe qubit and $n$ maximally mixed register qubits, $\rho_0 = |0\rangle\langle 0| \otimes I/2^n$, and applying $U = H_1 V_c^\dagger(g_{x_2}) V_c(g_{x_1}) H_1$ with controlled feature maps, the probe's expectation value $\langle \sigma_z \rangle = \mathrm{Re}[\mathrm{Tr}(V(g_{x_1})V^\dagger(g_{x_2}))]/2^n$ returns the normalized Hilbert–Schmidt inner product between the two embedding unitaries in a single readout. Substituted into the NQE loss, that number makes the whole pipeline—neural network, feature map, and loss—differentiable with respect to the network weights $w$, so ordinary gradient descent can learn the embedding. Two supporting facts carry the argument: trace distance is contractive under CPTP maps, so the embedding is the only stage where class separation can be created, and the NMR platform initializes the required one-clean-qubit state with gate fidelities above 99.5%, making the experimental overlap estimates trustworthy.

What would settle it

A decisive ablation: take the trained network outputs $g(x,w)$ for the 500 images and classify them with a classical linear or two-layer model, bypassing the quantum feature map and the parameterized circuit entirely. If that classical pipeline already reaches roughly 98% accuracy, the improvement is attributable to classical preprocessing rather than to the quantum embedding; if it stays near 54%, the separation is genuinely created by the embedding. A second consistency check is to compute the exact trace distance $\|\rho_+ - \rho_-\|_1$ between the class-conditional states after each NQE iteration and compare it with the NQE loss; if the loss decreases while the trace distance saturates or drops, the surrogate objective is not tracking the separation it claims to maximize.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the separation of quantum-embedded data can be optimized end-to-end with only one clean qubit. The NQE-DQC1 protocol composes a classical neural network $g(x,w)$ with the $ZZ$-feature map $V(\phi) = (\exp[i\sum_k \phi_k Z_k + \phi_{n+k}Z_kZ_{k+1}] H^{\otimes n})^M$ at $M=1$, so the network's outputs set the feature-map angles. The training signal is the normalized Hilbert–Schmidt inner product $\langle V(g_{x_i}), V(g_{x_j})\rangle_{HS} = \mathrm{Tr}(V(g_{x_i})V^\dagger(g_{x_j}))/2^n$, which DQC1 retrieves as the $z$-expectation value of a single probe qubit prepared in $|0\rangle\langle 0| \otimes I/2^n$. Minimizing the squared deviation of that overlap from 1 for same-class pairs and 0 for different-class pairs is the protocol's objective, and the paper takes this to maximize the trace distance between the class-averaged states $\rho_+$ and $\rho_-$; because any later quantum operation is a contractive map, that distance bounds every downstream classifier. Experimentally the loss converges around the tenth iteration, the measured trace distance rises on both training and test images, and a small two-layer parameterized circuit reaches 98% accuracy with the trained embedding versus 54% with the raw feature map, with the same trained network also guiding classification on a superconducting processor.

Load-bearing premise

The load-bearing premise is that matching the average overlap of whole embedding circuits to the label pattern—the quantity DQC1 measures with one clean qubit—faithfully maximizes how far apart the two classes of embedded quantum states actually end up, and that the large accuracy gain comes from the quantum embedding rather than from the classical neural network's preprocessing of the images.

Editorial extensions

If this is right

If the protocol is right, ensemble quantum systems such as NMR become a practical training resource for quantum machine learning: the DQC1 training loop needs only one clean qubit and a single readout per overlap estimate, which is exactly the regime these platforms can support.
A fixed, hand-picked feature map can be the binding constraint on a quantum classifier: with the same shallow parameterized circuit, accuracy on the MNIST 0/1 task jumps from 54% to 98% purely by replacing the raw feature map with the trained one.
The trained embedding transfers across hardware: the neural network weights learned on the NMR processor were reused to run classification on a cloud superconducting processor, with results tracking numerical simulation.
The protocol generalizes beyond the demonstration: supplementary runs on Fashion-MNIST and satellite imagery, under simulated NMR noise and on a 127-qubit superconducting processor, show the same qualitative advantage of NQE embedding over the raw $ZZ$-feature map.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural control the paper does not run: feed the trained network outputs $g(x,w)$ directly to a classical classifier, skipping the quantum feature map and PQC. If those features already separate the 500 images at near 98%, part of the reported gain is classical preprocessing; if they stay near 54%, the separation truly happens in the embedding.
The NQE loss is an indirect objective—it matches the average overlap of whole embedding circuits to the label pattern rather than the fidelity of the specific embedded states used at inference. A direct test would train against $\|\rho_+ - \rho_-\|_1$ itself on a small register and compare the trace distance actually achieved.
The paper demonstrates transfer in one direction, from NMR-trained embeddings to a superconducting processor; the complementary direction—training on a platform with higher gate fidelities and transferring back—is untested, though nothing in the protocol forbids it.
Because the single-readout overlap estimate is independent of the number of register qubits, the training signal's cost scales with the complexity of implementing the controlled feature maps rather than with register size—a structural hint that this training scheme, if correct, has room to grow beyond three encoding qubits.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The useful new thing here is an experimental demonstration that DQC1 can train a neural network to produce a quantum embedding on an NMR processor, and that the resulting embedding transfers to IBM superconducting hardware. The NQE concept and the Hilbert-Schmidt loss come from the group's own earlier work (Ref [14]); the DQC1-specific training loop, the NMR implementation, and the cross-platform transfer are the new pieces. The experiment looks self-consistent: the loss converges, trace distance increases for both training and test pairs, and the IBM results match the simulations.

The main soft spot is the headline comparison. The 98%-versus-54% classification result changes two things at once: the NQE pipeline inserts a trainable classical neural network before the ZZ feature map, while the baseline uses raw PCA features. So the comparison does not isolate the contribution of the DQC1-training objective. A classical neural network with the same architecture, trained with a standard loss on the same five PCA components, could plausibly produce comparable separation on MNIST 0/1, making the quantum embedding a near-trivial readout. Without an ablation that holds the NN architecture fixed and trains it with a classical loss, or at least reports the class separation before quantum encoding, the 98% number cannot be attributed to the NQE-DQC1 mechanism. This is the primary threat to the claim, and it is addressable.

The reader's worry that the loss is misaligned with trace distance is weaker than it first appears. For the implemented one-layer ZZ feature map, V(gx) = D(gx)H^⊗n, so the real part of the normalized Hilbert-Schmidt inner product equals the real part of the state fidelity. Optimizing toward 0/1 targets does push trace distance in the right direction. Minor caveat: the loss uses only the real part, leaving the imaginary part of the fidelity unconstrained, but the experimental trace-distance curves suggest this is not a practical issue here.

The data split for the 98% accuracy is ambiguous; it is not clear whether the 500 images are the training set, the test set, or the union. The authors should state this explicitly. That is a minor reporting issue.

This is a solid experimental methods paper for the QML-on-NMR subfield and deserves a serious referee. The referee should ask for the missing classical-NN baseline and a clear statement of the train/test split before the 98%-versus-54% claim is taken at face value.

Referee Report

4 major / 4 minor

Summary. The paper proposes Neural Quantum Embedding via DQC1 (NQE-DQC1), a hybrid classical-quantum method for learning quantum feature maps for binary classification. A classical neural network g(x,w) maps input data to parameters of a ZZ feature map, and the network is trained by minimizing Eq. (1), a squared deviation between the normalized Hilbert-Schmidt inner product of the embedding unitaries and a label-matching target. The HS inner product is estimated experimentally with a DQC1 circuit on a four-qubit NMR processor. The authors report that training loss decreases, the trace distance between class-conditional embedded states increases, and a subsequent parameterized quantum circuit achieves 98% classification accuracy on 500 MNIST 0/1 images, versus 54% with a traditional ZZ feature map. They also transfer the trained embedding to IBM superconducting processors and include supplementary experiments on Fashion-MNIST and satellite images.

Significance. If the central claims hold, the work would be a useful demonstration that a subuniversal DQC1 model can train a classical network-to-quantum embedding on an ensemble quantum processor and that the trained embedding transfers to other hardware. The experimental execution appears careful: the DQC1 trace-estimation circuit is standard, the measured training loss tracks simulation, and the IBM transfer experiment adds credibility. The paper also ships numerical and experimental loss curves for a second dataset in the supplement. However, the headline accuracy comparison conflates the quantum embedding with classical preprocessing, and the relationship between the optimized loss and the stated trace-distance objective is not derived for the general case. These issues must be resolved before the main claims can be accepted.

major comments (4)

[§Protocol overview, Eq. (1)] The text states that Eq. (1) implements the goal of maximizing the trace distance between the class-conditional states rho+ and rho-, but the loss actually minimizes the squared deviation of normalized Hilbert-Schmidt inner products of the embedding unitaries from the label-matching target. The equivalence between these two objectives is asserted, not derived. For the specific single-layer ZZ feature map used here (M=1), V(gx)=P(gx)H^⊗n, so the normalized HS inner product coincides with the fidelity between the embedded states, which partly closes the gap in the implemented experiment. Nevertheless, the general claim in the protocol overview and the discussion of CPTP-map contractivity should be accompanied by a derivation or an explicit condition under which Eq. (1) is equivalent to maximizing trace distance.
[§Classification results, Fig. 4] The central 98%-versus-54% comparison changes two variables at once: the NQE condition prepends the trained classical neural network g(x,w) to the ZZ feature map, while the 'traditional' baseline feeds raw PCA features directly into the same feature map. Because MNIST 0/1 with five PCA components is already a very easy binary problem, the improved separation and accuracy could be caused entirely by the classical neural-network preprocessing, with no contribution from the DQC1-based training objective. To support the headline claim, the authors should include an ablation in which the same neural network architecture is trained with a purely classical loss (or used with random weights) and then fed into the same quantum embedding and PQC; the measured accuracy in that condition would isolate the effect of the NQE-DQC1 training.
[§Classification results, Fig. 4(b)] The reported 98.0% accuracy is computed over 'all 500 images' with no separate training/test breakdown, even though the dataset is partitioned into training and test sets and Fig. 4(a) shows only eight images per set. Generalization is a central claim of the paper, so the authors should report test-set accuracy separately, along with the number of PQC training iterations, the batch selection procedure, and the variance over repeated PQC training runs. Without these, the reader cannot distinguish memorization of the training set from genuine separation.
[§Experimental scheme, NQE training] The neural network g(x,w) is never defined: no layer count, width, activation function, initialization, optimizer, or learning-rate schedule is given, and the paper does not state how ∇L_NQE is obtained from the DQC1-estimated loss. Since g is the trainable component of the protocol and the confounded-baseline question depends on its capacity, this omission prevents reproduction and makes it impossible to assess whether the network alone solves the classification task.

minor comments (4)

[Throughout] There are several typos and inconsistent references: 'nuclear magnatic resonance' on page 2, 'sturcture' on page 4, and 'Eq. (B1)' in the main text should refer to Eq. (1).
[§Experimental scheme] The PCA preprocessing is under-specified: the input dimensionality before PCA, whether PCA is fit on the training set only, and the normalization applied to the five PCA components should be stated.
[§Fig. 2(c)] The trace-distance error bars are standard deviations over 20 randomly chosen pairs; please clarify whether the 20-pair selection was fixed across iterations and whether multiple independent NQE training runs were performed, since the reported curves otherwise reflect a single training trajectory.
[§Supplemental Material, Appendix E] For the Fashion-MNIST and satellite datasets, only PQC loss curves are reported; end-task classification accuracies on held-out test data should be reported to support the claim that NQE improves these additional benchmarks.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the training loss, the trace-distance evaluation, and the PQC classification benchmark are distinct quantities, and the self-citations are not load-bearing.

full rationale

The paper's derivation chain is: (i) define the NQE loss in Eq. (1) as the squared deviation between the Hilbert-Schmidt inner product of embedding unitaries and the label-matching target; (ii) estimate this inner product on the NMR DQC1 circuit via <sigma_z> = Re[Tr(V(gx1)Vdagger(gx2))]/2^n; (iii) update the classical neural network by gradient descent on that loss; and (iv) evaluate the learned embedding with a separately trained parameterized quantum circuit. The training loss in Fig. 2(b) is literally the quantity being minimized, so its decrease is an optimization result, not a prediction. The claimed performance metrics are not equal to the training objective by construction: the trace distance in Fig. 2(c) is computed from the post-training embedded states, and the 98% versus 54% classification accuracy comes from a separately optimized PQC and is compared against the standard ZZ-feature embedding. The paper does not fit the reported accuracy into Eq. (1); the accuracy is an external benchmark. The self-citations, most notably Refs. [14] and [15], introduce the prior NQE framework and margin-based generalization analysis, but the DQC1 adaptation, the NMR implementation, and the transfer to IBM hardware are new contributions that do not reduce to those citations. One may criticize that the NQE-vs-baseline comparison changes both the classical neural preprocessing and the quantum trace-distance objective, and one may note that Eq. (1) is a surrogate for trace distance rather than an established equivalent; however, those are experimental-confound and correctness-risk concerns, not circularity. No equation in the paper is equivalent to its own input by construction, and no fitted parameter is presented as an independent prediction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. Its free parameters are the trained neural network weights and hand-chosen hyperparameters (PCA dimension, feature map layers, PQC angles). The axioms are standard DQC1 and trace-distance results plus the unproven surrogate assumption that the Hilbert-Schmidt loss drives trace distance separation.

free parameters (4)

neural network weights w = not specified (trained over 15 iterations)
The central claim depends on the trained NN that maps images to feature-map parameters; the NN architecture and hyperparameters are not given in the main text.
PCA dimension (5 components) = 5
Images are reduced to 1x5 vectors by PCA; the number of components is a hand-chosen hyperparameter that limits the feature-map size.
ZZ feature map layers M = 1
The number of layers in the embedding circuit is set to 1, a choice affecting expressivity.
PQC parameters theta = not specified (trained)
The four rotation parameters of the classifier circuit are trained with the parameter-shift rule; their final values are not reported.

assumptions (4)

standard math DQC1 can efficiently estimate the normalized trace of an n-qubit unitary, up to additive error, with bounded variance.
Invoked in Eq. (1) and the circuit in Fig. 2(a); from Knill-Laflamme and Nielsen-Chuang.
standard math Trace distance contracts under completely positive trace-preserving maps, so a CPTP map cannot increase separability of the class ensembles.
Used in SM Appendix B to argue the embedding is the only place to improve separability.
domain assumption A classical neural network trained to match Hilbert-Schmidt inner products to label targets will produce feature-map parameters that improve classification of the embedded quantum states.
The core heuristic; not proven, and the loss is a proxy rather than the trace distance itself.
domain assumption The ZZ feature map with M=1 and 5 features is expressive enough for the learned embedding on MNIST 0/1.
The experimental design assumes this circuit family can represent the needed embedding after NN preprocessing.

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Cite this review

Pith. "Pith review of Neural quantum embedding via deterministic quantum computation with one qubit." pith.science (2026). https://pith.science/paper/ZVWKKPKY

@misc{pith2026250115359,
  author       = {Pith},
  title        = {Pith review of: Neural quantum embedding via deterministic quantum computation with one qubit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZVWKKPKY}},
  note         = {Machine review of arXiv:2501.15359}
}

read the original abstract

Quantum computing is expected to provide exponential speedup in machine learning. However, optimizing the data loading process, commonly referred to as quantum data embedding, to maximize classification performance remains a critical challenge. In this work, we propose a neural quantum embedding (NQE) technique based on deterministic quantum computation with one qubit (DQC1). Unlike the traditional embedding approach, NQE trains a neural network to maximize the trace distance between quantum states corresponding to different categories of classical data. Furthermore, training is efficiently achieved using DQC1, which is specifically designed for ensemble quantum systems, such as nuclear magnetic resonance (NMR). We validate the NQE-DQC1 protocol by encoding handwritten images into NMR quantum processors, demonstrating a significant improvement in distinguishability compared to traditional methods. Additionally, after training the NQE, we implement a parameterized quantum circuit for classification tasks, achieving 98\% classification accuracy, in contrast to the 54\% accuracy obtained using traditional embedding. Moreover, we show that the NQE-DQC1 protocol is extendable, enabling the use of the NMR system for NQE training due to its high compatibility with DQC1, while subsequent machine learning tasks can be performed on other physical platforms, such as superconducting circuits. Our work opens new avenues for utilizing ensemble quantum systems for efficient classical data embedding into quantum registers.

Figures

Figures reproduced from arXiv: 2501.15359 by the authors.

**Figure 2.** FIG. 2. (a) Experimental quantum circuit for NQE-DQC1 training. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Quantum circuit for classification tasks. Classical images [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. Experimental classification results after processing through the PQC circuit for [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5. Handwritten images of “0” and “1” from the MNIST dataset that are used for our classification experiments. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. (a) Molecular structure and key parameters of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. The top and bottom panels present the results for [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8. Optimization results of the parameterized quantum circuit for the Fashion-MNIST (left) and satellite image (right) datasets. The [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

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Reference graph

Works this paper leans on

60 extracted references · 43 canonical work pages

[14]

T. Hur, I. F. Araujo, and D. K. Park, Neural quantum embedding: Pushing the limits of quantum supervised learning, Phys. Rev. A 110, 022411 (2024)

2024
[1]

Biamonte, P

J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature 549, 195 (2017)

2017
[2]

Rebentrost, M

P. Rebentrost, M. Mohseni, and S. Lloyd, Quantum support vector machine for big data classification, Phys. Rev. Lett.113, 130503 (2014)

work page 2014
[3]

Lloyd, M

S. Lloyd, M. Mohseni, and P. Rebentrost, Quantum principal component analysis, Nat. Phys.10, 631 (2014)

work page 2014
[4]

Cerezo, G

M. Cerezo, G. Verdon, H.-Y. Huang, L. Cincio, and P. J. Coles, Challenges and opportunities in quantum machine learning, Nat. Comput. Sci. 2, 567 (2022)

work page 2022
[5]

Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018)

J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018)

2018
[6]

Havl´ıcek, A

V. Havl´ıcek, A. D. C´orcoles, K. Temme, A. W. Harrow, A. Kan- dala, J. M. Chow, and J. M. Gambetta, Supervised learning with quantum-enhanced feature spaces, Nature 567, 209 (2019)

work page 2019
[7]

Benedetti, E

M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, Parame- terized quantum circuits as machine learning models, Quantum Sci. Technol. 4, 043001 (2019)

work page 2019

Show all 60 references

[8]

Cerezo, A

M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021)

2021
[9]

Bharti, A

K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin- Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru- Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022)

2022
[10]

Knill and R

E. Knill and R. Laflamme, Power of one bit of quantum infor- mation, Phys. Rev. Lett.81, 5672 (1998)

1998
[11]

Z. Li, X. Liu, N. Xu, and J. Du, Experimental realization of a quantum support vector machine, Phys. Rev. Lett. 114, 140504 (2015)

2015
[12]

T. Xin, L. Che, C. Xi, A. Singh, X. Nie, J. Li, Y. Dong, and D. Lu, Experimental quantum principal component analysis via parametrized quantum circuits, Phys. Rev. Lett. 126, 110502 (2021)

2021
[13]

P. L. Bartlett, D. J. Foster, and M. J. Telgarsky, Spectrally- normalized margin bounds for neural networks, Advances in neural information processing systems 30 (2017)

2017
[15]

Hur and D

T. Hur and D. K. Park, Understanding generalization in quantum machine learning with margins, arXiv preprint arXiv:2411.06919 (2024)

2024
[16]

Datta, S

A. Datta, S. T. Flammia, and C. M. Caves, Entanglement and the power of one qubit, Phys. Rev. A72, 042316 (2005)

2005
[17]

Poulin, R

D. Poulin, R. Blume-Kohout, R. Laflamme, and H. Ollivier, Exponential speedup with a single bit of quantum information: Measuring the average fidelity decay, Phys. Rev. Lett.92, 177906 (2004)

2004
[18]

P. W. Shor and S. P. Jordan, Estimating jones polynomials is a complete problem for one clean qubit, Quantum Info. Comput. 8, 681 (2008)

2008
[19]

D. K. Park, J.-K. K. Rhee, and S. Lee, Noise-tolerant parity learning with one quantum bit, Phys. Rev. A97, 032327 (2018)

2018
[20]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition , 10th ed. (Cambridge University Press, New York, NY, USA, 2011)

2011
[21]

D. G. Cory, A. F. Fahmy, and T. F. Havel, Ensemble quantum computing by nmr spectroscopy, Proc. Nat. Acad. Sci. U.S.A. 94, 1634 (1997)

1997
[22]

Criger, G

B. Criger, G. Passante, D. Park, and R. Laflamme, Recent ad- vances in nuclear magnetic resonance quantum information pro- cessing, Phil. Trans. R. Soc. A 370, 4620 (2012)

2012
[23]

J. Li, Z. Luo, T. Xin, H. Wang, D. Kribs, D. Lu, B. Zeng, and R. Laflamme, Experimental implementation of efficient quan- tum pseudorandomness on a 12-spin system, Phys. Rev. Lett. 123, 030502 (2019)

2019
[24]

thesis (2015)

Park, Kyungdeock, Coherent control of nuclear and electron spins for quantum information processing, Ph.D. thesis (2015)

2015
[25]

D. Lu, A. Brodutch, J. Park, H. Katiyar, T. Jochym-O’Connor, and R. Laflamme, Nmr quantum information processing, in Electron Spin Resonance (ESR) Based Quantum Computing , 6 edited by T. Takui, L. Berliner, and G. Hanson (Springer New York, New York, NY, 2016) pp. 193–226

2016
[26]

C. Qiu, X. Nie, and D. Lu, Quantum simulations with nuclear magnetic resonance system, Chin. Phys. B 30, 048201 (2021)

2021
[27]

Bae and L.-C

J. Bae and L.-C. Kwek, Quantum state discrimination and its applications, J. Phys. A: Math. Theor. 48, 083001 (2015)

2015
[28]

Grant, M

E. Grant, M. Benedetti, S. Cao, A. Hallam, J. Lockhart, V. Sto- jevic, A. G. Green, and S. Severini, Hierarchical quantum clas- sifiers, npj Quantum Inf. 4, 65 (2018)

2018
[29]

I. Cong, S. Choi, and M. D. Lukin, Quantum convolutional neural networks, Nat. Phys. 15, 1273 (2019)

2019
[30]

T. Hur, L. Kim, and D. K. Park, Quantum convolutional neural network for classical data classification, Quantum Mach. Intell. 4, 3 (2022)

2022
[31]

G. Park, J. Huh, and D. K. Park, Variational quantum one- class classifier, Machine Learning: Science and Technology 4, 015006 (2023)

2023
[32]

J. Kim, J. Huh, and D. K. Park, Classical-to-quantum convolu- tional neural network transfer learning, Neurocomputing 555, 126643 (2023)

2023
[33]

P ´erez-Salinas, A

A. P ´erez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre, Data re-uploading for a universal quantum classifier, Quantum 4, 226 (2020)

2020
[34]

Kim and D

Y. Kim and D. K. Park, Expressivity of deterministic quantum computation with one qubit, arXiv preprint arXiv:2411.02751 (2024)

2024 arXiv
[35]

Y. Liu, S. Arunachalam, and K. Temme, A rigorous and ro- bust quantum speed-up in supervised machine learning, Nature Physics 17, 1013 (2021)

2021
[36]

Huang, M

H.-Y. Huang, M. Broughton, M. Mohseni, R. Babbush, S. Boixo, H. Neven, and J. R. McClean, Power of data in quantum machine learning, Nat. Commun. 12, 2631 (2021)

2021
[37]

Siudzi ´nska, S

K. Siudzi ´nska, S. Chakraborty, and D. Chru ´sci´nski, Interpo- lating between positive and completely positive maps: a new hierarchy of entangled states, Entropy 23, 625 (2021)

2021
[38]

See Supplemental Material for more details
[39]

LeCun, C

Y. LeCun, C. Cortes, and C. Burges, Mnist hand- written digit database, ATT Labs [Online]. Available: http://yann.lecun.com/exdb/mnist 2 (2010)

2010
[40]

Long, W.-T

X. Long, W.-T. He, N.-N. Zhang, K. Tang, Z. Lin, H. Liu, X. Nie, G. Feng, J. Li, T. Xin, Q. Ai, and D. Lu, Entanglement- enhanced quantum metrology in colored noise by quantum zeno effect, Phys. Rev. Lett.129, 070502 (2022)

2022
[41]

Huang, C

K. Huang, C. Xi, X. Long, H. Liu, Y.-a. Fan, X. Wang, Y. Zheng, Y. Feng, X. Nie, and D. Lu, Experimental realization of self- contained quantum refrigeration, Phys. Rev. Lett. 132, 210403 (2024)

2024
[42]

C. Xi, X. Liu, H. Liu, K. Huang, X. Long, D. Ebler, X. Nie, O. Dahlsten, and D. Lu, Experimental validation of enhanced information capacity by quantum switch in accordance with thermodynamic laws, Phys. Rev. Lett.133, 040401 (2024)

2024
[43]

Z. Lin, H. Liu, K. Tang, Y. Liu, L. Che, X. Long, X. Wang, Y.- a. Fan, K. Huang, X. Yang, et al., Hardware-efficient quantum principal component analysis for medical image recognition, Front. Phys. 19, 51202 (2024)

2024
[44]

Khaneja, T

N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr¨ uggen, and S. J. Glaser, Optimal control of coupled spin dynamics: design of nmr pulse sequences by gradient ascent algorithms, J. Magn. Reson. 172, 296 (2005)

2005
[45]

D. G. Cory, M. D. Price, and T. F. Havel, Nuclear magnetic res- onance spectroscopy: An experimentally accessible paradigm for quantum computing, Physica D 120, 82 (1998)

1998
[46]

J. Li, X. Yang, X. Peng, and C.-P. Sun, Hybrid quantum-classical approach to quantum optimal control, Phys. Rev. Lett. 118, 150503 (2017)

2017
[47]

D. Lu, K. Li, J. Li, H. Katiyar, A. J. Park, G. Feng, T. Xin, H. Li, G. Long, A. Brodutch, et al., Enhancing quantum control by bootstrapping a quantum processor of 12 qubits, npj Quantum Inf. 3, 45 (2017)

2017
[48]

G. Feng, F. H. Cho, H. Katiyar, J. Li, D. Lu, J. Baugh, and R. Laflamme, Gradient-based closed-loop quantum optimal con- trol in a solid-state two-qubit system, Phys. Rev. A 98, 052341 (2018)

2018
[49]

X. Lin, J. Fan, R. Ye, M. Zhou, Y. Song, D. Lu, and N. Xu, Online optimization for optical readout of a single electron spin in diamond, Front. Phys. 18, 21301 (2023)

2023
[50]

M. M. Wilde, Quantum Information Theory (Cambridge Uni- versity Press, 2013)

2013
[51]

Lloyd, M

S. Lloyd, M. Schuld, A. Ijaz, J. Izaac, and N. Killoran, Quantum embeddings for machine learning, arXiv preprint arXiv:2001.03622 (2020)

2020 arXiv
[52]

J. R. Glick, T. P. Gujarati, A. D. Corcoles, Y. Kim, A. Kandala, J. M. Gambetta, and K. Temme, Covariant quantum kernels for data with group structure, arXiv preprint arXiv:2105.03406 (2021)

2021 arXiv
[53]

Hubregtsen, D

T. Hubregtsen, D. Wierichs, E. Gil-Fuster, P.-J. H. S. Derks, P. K. Faehrmann, and J. J. Meyer, Training quantum embedding kernels on near-term quantum computers, Phys. Rev. A 106, 042431 (2022)

2022
[54]

J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 4812 (2018), 1803.11173

2018 arXiv
[55]

Thanasilp, S

S. Thanasilp, S. Wang, M. Cerezo, and Z. Holmes, Exponential concentration and untrainability in quantum kernel methods, arXiv preprint arXiv:2208.11060 (2022)

2022 arXiv
[56]

X. Nie, X. Zhu, K. Huang, K. Tang, X. Long, Z. Lin, Y. Tian, C. Qiu, C. Xi, X. Yang, et al., Experimental realization of a quantum refrigerator driven by indefinite causal orders, Phys. Rev. Lett.129, 100603 (2022)

2022
[57]

A. J. Park, E. McKay, D. Lu, and R. Laflamme, Simulation of anyonic statistics and its topological path independence using a seven-qubit quantum simulator, New J. Phys.18, 043043 (2016)

2016
[58]

X. Nie, X. Zhu, Y.-a. Fan, X. Long, H. Liu, K. Huang, C. Xi, L. Che, Y. Zheng, Y. Feng, et al., Self-consistent determina- tion of single-impurity anderson model using hybrid quantum- classical approach on a spin quantum simulator, Phys. Rev. Lett. 133, 140602 (2024)

2024
[59]

Zhang, X

Z. Zhang, X. Long, X. Zhao, Z. Lin, K. Tang, H. Liu, X. Yang, X. Nie, J. Wu, J. Li, et al., Identifying abelian and non-abelian topological orders in the string-net model using a quantum scat- tering circuit, Phys. Rev. A105, L030402 (2022)

2022
[60]

0” and “1

H. Xiao, K. Rasul, and R. Vollgraf, Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms (2017), cs.LG/1708.07747. 7 Supplemental Material: Neural quantum embedding via deterministic quantum computation with one qubit Hongfeng Liu1, Tak Hur2, Shita...

2017 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [14]

T. Hur, I. F. Araujo, and D. K. Park, Neural quantum embedding: Pushing the limits of quantum supervised learning, Phys. Rev. A 110, 022411 (2024)

2024

[2] [1]

Biamonte, P

J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature 549, 195 (2017)

2017

[3] [2]

Rebentrost, M

P. Rebentrost, M. Mohseni, and S. Lloyd, Quantum support vector machine for big data classification, Phys. Rev. Lett.113, 130503 (2014)

work page 2014

[4] [3]

Lloyd, M

S. Lloyd, M. Mohseni, and P. Rebentrost, Quantum principal component analysis, Nat. Phys.10, 631 (2014)

work page 2014

[5] [4]

Cerezo, G

M. Cerezo, G. Verdon, H.-Y. Huang, L. Cincio, and P. J. Coles, Challenges and opportunities in quantum machine learning, Nat. Comput. Sci. 2, 567 (2022)

work page 2022

[6] [5]

Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018)

J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018)

2018

[7] [6]

Havl´ıcek, A

V. Havl´ıcek, A. D. C´orcoles, K. Temme, A. W. Harrow, A. Kan- dala, J. M. Chow, and J. M. Gambetta, Supervised learning with quantum-enhanced feature spaces, Nature 567, 209 (2019)

work page 2019

[8] [7]

Benedetti, E

M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, Parame- terized quantum circuits as machine learning models, Quantum Sci. Technol. 4, 043001 (2019)

work page 2019